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A study on the mild solution of impulsive fractional evolution equations

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  • shu, Xiao-Bao
  • Shi, Yajing

Abstract

This paper is concerned with the formula of mild solutions to impulsive fractional evolution equation. For linear fractional impulsive evolution equations [8–25,27,30,31], described mild solution as integrals over (tk,tk+1](k=1,2,…,m) and [0, t1]. On the other hand, in [26,28,29], their solutions were expressed as integrals over [0, t]. However, it is still not clear what are the correct expressions of solutions to the fractional order impulsive evolution equations. In this paper, firstly, we prove that the solutions obtained in [8–25,27,30,31] are not correct; secondly, we present the right form of the solutions to linear fractional impulsive evolution equations with order 0 < α < 1 and 1 < α < 2, respectively; finally, we show that the reason that the solutions to an impulsive ordinary evolution equation are not distinct.

Suggested Citation

  • shu, Xiao-Bao & Shi, Yajing, 2016. "A study on the mild solution of impulsive fractional evolution equations," Applied Mathematics and Computation, Elsevier, vol. 273(C), pages 465-476.
    • Handle: RePEc:eee:apmaco:v:273:y:2016:i:c:p:465-476
    DOI: 10.1016/j.amc.2015.10.020
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    References listed on IDEAS

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    1. Zhenhai Liu & Xiuwen Li, 2013. "On the Controllability of Impulsive Fractional Evolution Inclusions in Banach Spaces," Journal of Optimization Theory and Applications, Springer, vol. 156(1), pages 167-182, January.
    2. JinRong Wang & Michal Fec̆kan & Yong Zhou, 2013. "Relaxed Controls for Nonlinear Fractional Impulsive Evolution Equations," Journal of Optimization Theory and Applications, Springer, vol. 156(1), pages 13-32, January.
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    Cited by:

    1. Sousa, J. Vanterler da C. & Oliveira, D.S. & Capelas de Oliveira, E., 2021. "A note on the mild solutions of Hilfer impulsive fractional differential equations," Chaos, Solitons & Fractals, Elsevier, vol. 147(C).
    2. Mohan Raja, M. & Vijayakumar, V., 2022. "Existence results for Caputo fractional mixed Volterra-Fredholm-type integrodifferential inclusions of order r ∈ (1,2) with sectorial operators," Chaos, Solitons & Fractals, Elsevier, vol. 159(C).
    3. Longfei Lin & Yansheng Liu & Daliang Zhao, 2021. "Controllability of Impulsive ψ -Caputo Fractional Evolution Equations with Nonlocal Conditions," Mathematics, MDPI, vol. 9(12), pages 1-14, June.

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